(1) Be found to exist;

(2) May seem not to exist, but nevertheless may really exist;

(3) May actually be non-existent.

In the second case the failure is only apparent, and arises from our obtuseness of perception, the smallness of the phenomenon to be noticed, or the disguised character in which it appears. I have already pointed out that the analogy of sound and light seems to fail because light does not bend round a corner, the fact being that it does so bend in the phenomena of diffraction, which present the effect, however, in such an unexpected and minute form, that even Newton was misled, and turned from the correct hypothesis of undulations which he had partially entertained.

In the third class of cases analogy fails us altogether, and we expect that to exist which really does not exist. Thus we fail to discover the phenomena of polarization in sound travelling through the atmosphere, since air is not capable of any appreciable transverse undulations. These failures of analogy are of peculiar interest, because they make the mind aware of its superior powers. There have been many philosophers who said that we can conceive nothing in the intellect which we have not previously received through the senses. This is true in the sense that we cannot image them to the mind in the concrete form of a shape or a colour; but we can speak of them and reason concerning them; in short, we often know them in everything but a sensuous manner. Accurate investigation shows that all material substances retard the motion of bodies through them by substracting energy by impact. By the law of continuity we can frame the notion of a vacuous space in which there is no resistance whatever, nor need we stop there; for we have only to proceed by analogy to the case where a medium should

accelerate the motion of bodies passing through it, somewhat in the mode which Aristotelians attributed falsely to the air. Thus we can frame the notion of negative density, and Newton could reason exactly concerning it, although no such thing exists 9.

In every direction of thought we may meet ultimately with similar failures of analogy. A moving point generates a line, a moving line generates a surface, a moving surface generates a solid, but what does a moving solid generate? When we compare a polyhedron, or manysided solid, with a polygon, or plane figure of many sides, the volume of the first is analogous to the area o the second; the face of the solid answers to the side of the polygon; the edge of the solid to the point of the figure; but the corner, or junction of edges in the polyhedron, is left wholly unrepresented in the plane of the polygon. Even if we attempted to draw the analogies in some other manner, we should still find a geometrical notion embodied in the solid which has no representative in the plain figurer.

Faraday was able to frame some notion of matter in a fourth condition, which should be to gas what gas is to liquids. Such substance, he thought, would not fall far short of radiant matter, by which apparently he meant the supposed caloric or matter assumed to constitute heat, according to the Corpuscular Theory. Even if we could frame the notion, matter in such a state cannot be known to exist, and recent discoveries concerning the continuity of the solid, liquid, and gaseous states remove the basis of the speculation.

From these and many other instances which might be

a 'Principia,' bk. II. Section II. Prop. X.

r De Morgan, Cambridge Philosophical Transactions,' vol. xi. Part ii. p. 246.

'Life of Faraday,' vol. i. p. 216.

adduced, we learn that analogical reasoning leads us to the conception of many things which, so far as we can ascertain, do not exist. In this way great perplexities have arisen in the use of language and mathematical symbols. All language depends upon analogy; for we join and arrange words so that they may represent the corresponding junctions or arrangements of things and their qualities. But in the use of language we are obviously capable of forming many combinations of words to which no corresponding meaning apparently exists. The same difficulty arises in the use of mathematical signs, and mathematicians have needlessly puzzled themselves about the square root of a negative quantity, which is, in many applications of algebraic calculation, simply a sign without any analogous meaning, there being a failure of analogy.

CHAPTER XXIX.

EXCEPTIONAL PHENOMENA.

IF science consists in the detection of identity and the recognition of one uniformity existing in many objects, it follows almost of necessity that the progress of science depends upon the study of exceptional phenomena. Such new phenomena are the raw material upon which we are to exert our faculties of observation and reasoning, in order to reduce the new facts beneath the sway of the laws of nature, either those laws already well known, or those to be discovered. Not only are strange and inexplicable facts those which are on the whole most likely to lead us to some novel and important discovery, but they are also best fitted to arouse our attention. So long as events happen in accordance with our anticipations, and the routine of every-day observation is unvaried, there is nothing to impress upon the mind the smallness of its knowledge, and the depth of mystery, which may be hidden in the commonest sights and objects. In early times the myriads of stars which remained in apparently fixed relative positions upon the heavenly sphere, received far less notice from astronomers than those few planets whose wandering and inexplicable motions formed an unsolved riddle. Hipparchus was induced to prepare the first catalogue of stars, because a single new star had been added to those nightly visible; and in the middle

ages two brilliant but temporary stars caused more popular interest in astronomy than any other events, and to one of them we owe all the observations of Tycho Brahe, the medieval Hipparchus.

In other sciences, as well as in that of the heavens, exceptional events are commonly the points from which we start to explore new regions of knowledge. It has been beautifully said that Wonder is the daughter of Ignorance, but the mother of Invention; and though the most familiar and slight events, if fully examined, will afford endless food for wonder and for wisdom, yet it is the few peculiar and unlooked-for events which most often lead a scientific mind into a course of discovery. It is true, indeed, that it requires much philosophy to observe things which are too near to us.

The high scientific importance attaching, then, to exceptions, renders it desirable that we should carefully consider the various modes in which an exception may be disposed of; while some new facts will be found to confirm the very laws to which at first sight they seem clearly opposed, others will cause us to limit the generality of our previous statements. In some cases the exception may be proved to be no exception; occasionally it will prove fatal to our previous most confident speculations; and there are some new phenomena which, without really destroying any of our former theories, open to us wholly new fields of scientific investigation. The study of this subject is especially interesting and important, because, as I have before said (vol. ii. p. 233), no important theory can be built up complete and perfect all at once. When unexplained phenomena present themselves as objections to the theory, it will often demand the utmost judgment and sagacity to assign to them their proper place and force. The acceptation or rejection of a theory will entirely depend upon discriminating the one insuperable contra